Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space
For a sectorial operator A with spectrum σ(A) that acts in a complex Banach space B, we prove that the condition σ(A) ∩ i R = Ø is sufficient for the differential equation \(\varepsilon x_\varepsilon^\prime\prime(t)+x_\varepsilon^\prime(t)=Ax_\varepsilon(t)+f(t), t \in R,\) where ε is a small positive parameter, to have a unique bounded solution x ε for an arbitrary bounded function f: R → B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as ε → 0+ to the unique bounded solution of the differential equation x′(t) = Ax(t) + f(t).
English version (Springer): Ukrainian Mathematical Journal 55 (2003), no. 7, pp 1071-1085.
Citation Example: Gorodnii M. F. Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space // Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 889-900.